We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$. We are interested in the sequence $\left( M ^ { k } \right) _ { k \in \mathbb { N } }$ of powers of $M$. We denote by $m _ { i , j } ^ { ( k ) }$ the coefficient of the matrix $M ^ { k }$ located in row $i$ and column $j$. For all $j \in \llbracket 1 , n \rrbracket$, we set $$\left\{ \begin{array} { l } \alpha _ { j } ^ { ( k ) } = \min _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } , \\ \beta _ { j } ^ { ( k ) } = \max _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } . \end{array} \right.$$ In the following four questions, $j$ is a fixed integer in $\llbracket 1 , n \rrbracket$ and $k$ is fixed in $\mathbb { N }$. Prove that there exists a pair $\left( i _ { 0 } , j _ { 0 } \right) \in \llbracket 1 , n \rrbracket ^ { 2 }$ such that $$\alpha _ { j } ^ { ( k + 1 ) } - \alpha _ { j } ^ { ( k ) } \geqslant m _ { i _ { 0 } , j _ { 0 } } \left( \beta _ { j } ^ { ( k ) } - \alpha _ { j } ^ { ( k ) } \right) .$$
We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$. We are interested in the sequence $\left( M ^ { k } \right) _ { k \in \mathbb { N } }$ of powers of $M$. We denote by $m _ { i , j } ^ { ( k ) }$ the coefficient of the matrix $M ^ { k }$ located in row $i$ and column $j$.
For all $j \in \llbracket 1 , n \rrbracket$, we set
$$\left\{ \begin{array} { l } \alpha _ { j } ^ { ( k ) } = \min _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } , \\ \beta _ { j } ^ { ( k ) } = \max _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } . \end{array} \right.$$
In the following four questions, $j$ is a fixed integer in $\llbracket 1 , n \rrbracket$ and $k$ is fixed in $\mathbb { N }$. Prove that there exists a pair $\left( i _ { 0 } , j _ { 0 } \right) \in \llbracket 1 , n \rrbracket ^ { 2 }$ such that
$$\alpha _ { j } ^ { ( k + 1 ) } - \alpha _ { j } ^ { ( k ) } \geqslant m _ { i _ { 0 } , j _ { 0 } } \left( \beta _ { j } ^ { ( k ) } - \alpha _ { j } ^ { ( k ) } \right) .$$